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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2402.02282 |
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| _version_ | 1866929234095636480 |
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| author | Gan, Aiping Guo, Li |
| author_facet | Gan, Aiping Guo, Li |
| contents | A differential operator of weight $λ$ is the algebraic abstraction of the difference quotient $d_λ(f)(x):=\big(f(x+λ)-f(x)\big)/λ$, including both the derivation as $λ$ approaches to $0$ and the difference operator when $λ=1$. Correspondingly, differential algebra of weight $λ$ extends the well-established theories of differential algebra and difference algebra. In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. We show that differential operators of weight $-1$ on a lattice coincide with differential operators, while differential operators are special cases of difference operators. Distributivity of a lattice is characterized by the existence of certain difference operators. Furthermore, we characterize and enumerate difference operators on finite chains and finite quasi-antichains. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_02282 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Difference operators on lattices Gan, Aiping Guo, Li Rings and Algebras Category Theory 06B20, 06D05, 12H10, 13N15, 05A15, 39A70 A differential operator of weight $λ$ is the algebraic abstraction of the difference quotient $d_λ(f)(x):=\big(f(x+λ)-f(x)\big)/λ$, including both the derivation as $λ$ approaches to $0$ and the difference operator when $λ=1$. Correspondingly, differential algebra of weight $λ$ extends the well-established theories of differential algebra and difference algebra. In this paper, we initiate the study of differential operators with weights, in particular difference operators, on lattices. We show that differential operators of weight $-1$ on a lattice coincide with differential operators, while differential operators are special cases of difference operators. Distributivity of a lattice is characterized by the existence of certain difference operators. Furthermore, we characterize and enumerate difference operators on finite chains and finite quasi-antichains. |
| title | Difference operators on lattices |
| topic | Rings and Algebras Category Theory 06B20, 06D05, 12H10, 13N15, 05A15, 39A70 |
| url | https://arxiv.org/abs/2402.02282 |